Jack Sarkissian, PhD Algostox Trading jack@algostox.com Quantum coupled-wave model of price formation Physics of Financial Markets Series Jack Sarkissian, PhD Algostox Trading jack@algostox.com © Algostox Trading LLC. All rights reserved.

Market maker’s problem Market makers (MM) provide liquidity and profit from bid-ask spread When MM’s quote is matched the MM forms a position (inventory) Inventory carries market risk (residual risk) MMs try to close the position at profit as soon as possible Understanding spread behavior is crucial to MMs Will allow to price securities when liquidity is essential Will allow to manage risks of illiquid securities © Algostox Trading LLC. All rights reserved.

© Algostox Trading LLC. All rights reserved. How does price form? Price formation There is a spectrum of attainable prices Price is localized between bid and ask levels Single value is selected from spectrum at the time of transaction Each transaction is an elementary act of price measurement Common points with two-level quantum systems Interaction and transfer between price levels Bandgap between bid and ask and noncrossing terms (Wigner-von- Neumann theorem) Fluctuation of coupling coefficients This points at possibility to apply quantum framework to price formation Order book Ask size Price Bid size 100 27.87   1000 27.90 27.95 300 28.15 28.20 27.83 27.82 400 27.80 200 27.79 1200 27.78 © Algostox Trading LLC. All rights reserved.

Quantum framework for price Security state is described by probability amplitude: 𝑝= 𝜓 2 States are eigenfunctions and prices are eigenvalues of the price operator: 𝑺 𝜓 𝑛 = 𝑠 𝑛 𝜓 𝑛 For two-level system 𝜓= 𝜓 𝑎𝑠𝑘 𝜓 𝑏𝑖𝑑 and 𝑠 11 𝑠 12 𝑠 12 ∗ 𝑠 22 𝜓 𝑎𝑠𝑘 𝜓 𝑏𝑖𝑑 = 𝑠 𝑎𝑠𝑘/𝑏𝑖𝑑 𝜓 𝑎𝑠𝑘 𝜓 𝑏𝑖𝑑 Price operator fluctuates in time: 𝑺 𝑡+𝛿𝑡 =𝑺 𝑡 +𝜹𝑺 © Algostox Trading LLC. All rights reserved.

© Algostox Trading LLC. All rights reserved. Spread as a bandgap Order book 𝑠 𝑎𝑠𝑘 = 𝑠 𝑚𝑖𝑑 + Δ 2 and 𝑠 𝑏𝑖𝑑 = 𝑠 𝑚𝑖𝑑 − Δ 2 𝑠 𝑚𝑖𝑑 = 𝑠 11 + 𝑠 22 2 = 𝑠 𝑏𝑖𝑑 + 𝑠 𝑎𝑠𝑘 2 mid−price Δ= 𝑠 11 − 𝑠 22 2 +4 𝑠 12 2 (spread or high−low) Ask size Price Bid size 100 27.87   1000 27.90 27.95 300 28.15 28.20 27.83 27.82 400 27.80 200 27.79 1200 27.78 © Algostox Trading LLC. All rights reserved.

Price simulation in coupled-wave model Spread parametrization 𝑠 11 𝑡+𝑑𝑡 = 𝑠 𝑚𝑖𝑑 𝑡 +𝜎𝑑𝑧+ 𝜉 2 , 𝑠 22 𝑡+𝑑𝑡 = 𝑠 𝑚𝑖𝑑 𝑡 +𝜎𝑑𝑧− 𝜉 2 , 𝑠 12 𝑡 = 𝜅 2 , 𝑑𝑧∼𝑁(0,1), 𝜉~𝑁( 𝜉 0 , 𝜉 1 ) and 𝜅~𝑁( 𝜅 0 , 𝜅 1 ). Coupled-wave model 𝑠 𝑚𝑖𝑑 𝑡+𝑑𝑡 = 𝑠 𝑙𝑎𝑠𝑡 𝑡 +𝜎𝑑𝑧 𝑠 ℎ𝑖𝑔ℎ = 𝑠 𝑚𝑖𝑑 + Δ 2 , 𝑠 𝑙𝑜𝑤 = 𝑠 𝑚𝑖𝑑 − Δ 2 , and Δ= 𝜉 2 + 𝜅 2 𝑠 𝑙𝑎𝑠𝑡 (𝑡+𝑑𝑡)~𝑢𝑛𝑖𝑓( 𝑠 𝑙𝑜𝑤 , 𝑠 ℎ𝑖𝑔ℎ ) or 𝑠 𝑙𝑎𝑠𝑡 (𝑡+𝑑𝑡)~𝑁 𝑠 𝑚𝑖𝑑 ,Δ © Algostox Trading LLC. All rights reserved.

Data sources for the model Order book: NBBO Order book: effective bid/ask OHLC bar time series © Algostox Trading LLC. All rights reserved.

Calibration to bid-ask spread AAPL AMZN Relative spread (bp) Ticker  ξ1 κ0 κ1 AAPL 1.1 2.3 1.6 AMZN 3.2 4.8 2.7 GOOG 4.1 3.3 2.1 INTC 4.2 5.5 0.7 MSFT 2.6 0.16 GAZP (MoEx) 0.2 3.5 1.7 GOOG GAZP © Algostox Trading LLC. All rights reserved.

Calibration to bar data AAPL AMZN High-low (relative) Ticker  ξ1 κ0 κ1 AAPL 1.75% 1.31% 0.51% AMZN 1.53% 1.49% 0.45% GOOG 1.24% 1.02% 0.34% INTC 1.39% 1.11% 0.31% MSFT 1.10% 0.39% GAZP (MoEx) 1.73% 1.30% 0.35% GOOG GAZP © Algostox Trading LLC. All rights reserved.

© Algostox Trading LLC. All rights reserved. Risk management Coupled-wave model also allows to manage risk of illiquid securities Risk sources: Uncertainty of mid-price Uncertainty of spread / high-low bar size © Algostox Trading LLC. All rights reserved.

© Algostox Trading LLC. All rights reserved. Conclusions Every financial transaction is an elementary act of price measurement Price is an eigenvalue of price operator with fluctuating matrix elements Bid and ask prices behave like quantum chaotic quantities Model can be calibrated to simulate bid-ask, effective bid/ask, or OHLC prices Provides capabilities for pricing illiquid securities and managing their risks © Algostox Trading LLC. All rights reserved.

© Algostox Trading LLC. All rights reserved. Notice Please cite as: J. Sarkissian, “Quantum coupled-wave model of price formation (Physics of Financial Markets Series)”, released by Algostox Trading, June 1, 2016. Available from www.algostox.com. Copyright © Algostox Trading LLC. All rights reserved. Federal law prohibits the unauthorized reproduction, distribution or exhibition of the materials. Violations of copyright law will be prosecuted. These materials are released for informational purposes only. No representation or warranty, express or implied, is made by Algostox Trading LLC as to the accuracy, completeness, or fitness for any particular purpose of enclosed materials. Under no circumstances shall Algostox Trading LLC have any liability to any other person or any entity for (a) any loss, damage or other injury in whole or in part caused by, resulting from or relating to, any error (negligent or otherwise), of Algostox Trading LLC in connection with the compilation, analysis, interpretation, communication, publication or delivery of enclosed materials, or (b) any direct, indirect, special, consequential, incidental or compensatory damages whatsoever (including, without limitation, lost profits), in either case caused by reliance upon or otherwise resulting from or relating to the use of (including the inability to use) enclosed materials. The enclosed materials do not constitute or imply any financial advice. No recommendation is being made to buy or sell any securities. © Algostox Trading LLC. All rights reserved.